Prepared for submission to JCAP
Cosmic-ray propagation with DRAGON2:
I. numerical solver and astrophysical
ingredients
Carmelo Evoli
aDaniele Gaggero
bAndrea Vittino
cGiuseppe Di
Bernardo
dMattia Di Mauro
eArianna Ligorini
fPiero Ullio
gDario
Grasso
haGran Sasso Science Institute, viale Francesco Crispi 7, 67100 L’Aquila (AQ), Italy
bGRAPPA Institute, University of Amsterdam, Science Park 904, 1090 GL Amsterdam, The Netherlands
cPhysik-Department T30d, Technische Universit¨at M¨unchen, James-Franck-Straße 1, D-85748 Garching, Germany
dMax-Planck-Institut f¨ur Astrophysik, Karl-Schwarzschild-Straße 1, 85740 Garching bei M¨unchen, Germany
eW. W. Hansen Experimental Physics Laboratory, Kavli Institute for Particle Astrophysics and Cosmology, Department of Physics and SLAC National Accelerator Laboratory, Stan-ford University, StanStan-ford, CA 94305, USA
fInstytut Fizyki J¸adrowej - PAN, ul. Radzikowskiego 152, 31-342 Krak´ow, Poland
gScuola Internazionale di Studi Superiori Avanzati, via Bonomea 265, 34136 Trieste, Italy hINFN and Dipartimento di Fisica “E. Fermi”, Pisa University, Largo B. Pontecorvo 3,
I-56127 Pisa, Italy
E-mail: carmelo.evoli@gssi.infn.it,d.gaggero@uva.nl,andrea.vittino@tum.de, bernardo@mpa-garching.mpg.de,mdimauro@slac.stanford.edu,
arianna.ligorini@ifj.edu.pl,piero.ullio@sissa.it,dario.grasso@pi.infn.it
Abstract. We present version 2 of the DRAGON code designed for computing realistic pre-dictions of the CR densities in the Galaxy. The code numerically solves the interstellar CR transport equation (including inhomogeneous and anisotropic diffusion, either in space and momentum, advective transport and energy losses), under realistic conditions.
The new version includes an updated numerical solver and several models for the as-trophysical ingredients involved in the transport equation. Improvements in the accuracy of the numerical solution are proved against analytical solutions and in reference diffusion scenarios.
The novel features implemented in the code allow to simulate the diverse scenarios proposed to reproduce the most recent measurements of local and diffuse CR fluxes, going beyond the limitations of the homogeneous galactic transport paradigm. To this end, several applications using DRAGON2 are presented as well.
This new version facilitates the users to include their own physical models by means of a modular C++ structure.
Keywords: galactic cosmic rays
Contents
1 Introduction 1
2 Transport of CRs in the Galaxy 3
3 Numerical solution of the transport equation 5
3.1 Discretization over grid 5
3.2 Iteration scheme 6
3.3 Local One Dimensional (LOD) operator splitting method 6
3.4 Crank-Nicolson coefficients 7
4 Validation of the numerical solver 8
4.1 Spatial diffusion 9
4.2 Advection 10
4.3 Momentum diffusion 11
4.4 Energy losses 12
4.5 Convergence and variable time step 12
5 Physical applications based on new features 15
5.1 Inhomogeneous diffusion coefficient: The gradient problem 17 5.2 Variable scaling of the diffusion coefficient 18
5.3 Anisotropic diffusion coefficient 19
5.4 Pion momentum losses 21
5.5 Anisotropic diffusion from a transient source 21
5.6 Non-equidistant binning 22
5.7 Interplay of diffusion, reacceleration, and leptonic energy losses: the role of
the boundary condition in momentum. 25
6 Conclusions 27
A CN Tables 29
B Analytical solutions to the transport equations 31
B.1 Two-dimensional inhomogeneous and anisotropic diffusion equation 31 B.2 Three-dimensional anisotropic diffusion equation 31
B.3 Energy-loss equation 32 B.4 Reacceleration equation 32 B.5 Advection equation 33 C Astrophysical ingredients 34 C.1 Gas distributions 34 C.1.1 HI gas distributions 34 C.1.2 H2 gas distributions 36 C.1.3 XCO models 37
C.1.4 HII gas models 37
C.2.1 Regular component 38
C.2.2 Random component 40
C.3 Interstellar radiation field 40
C.4 Source term 42 C.4.1 Source profile 42 C.4.2 Source spectra 43 C.4.3 Source normalization 44 C.5 Wind velocity 44 C.6 Alfv´en velocity 45
C.7 Spiral galactic pattern 45
C.8 Spatial diffusion coefficient 46
C.9 Momentum diffusion coefficient 47
C.10 Energy losses 48
C.10.1 Synchrotron radiation 48
C.10.2 Inverse Compton Scattering 48
C.10.3 Bremsstrahlung 50
C.10.4 Ionisation Losses 50
C.10.5 Coulomb scattering 51
C.10.6 Pion production 53
D Notations in this paper 54
1 Introduction
An impressive experimental effort has been devoted during the last years to provide very precise measurements of the high-energy cosmic radiation.
Since 2006 the PAMELA [1] orbital observatory has measured the spectra of many charged cosmic-ray (CR) species and discovered several intriguing anomalies in the proton, helium and positron spectrum [2–4]; AMS-02 [5], on board of the International Space Station since 2011, confirmed some of those results with higher accuracy extending PAMELA mea-surements up to the TeV [6–9]. At even larger energies CALET [10] and ISS-CREAM [11] should soon bridge direct measurements with those of ground based air-shower experiments, like KASCADE-Grande [12], and ARGO-YBG [13].
Concerning gamma-rays, the Large Area Telescope (LAT) on board of the Fermi Gamma-Ray Space Telescope [14] has been releasing refined all-sky maps – as well as detailed imaging and spectral information of very interesting regions like the Galactic Center (GC) [15] and of extended sources like Supernova Remnants (SNRs) [16] – up to TeV energies [17], tracing the CR emission through the whole Galaxy. HAWC [18] and CTA [19] should soon extend those measurements to higher energy. At the opposite edge of the electromagnetic spectrum, in the radio and microwave bands, PLANCK [20], LOFAR [21] – and in the next future SKA [22] – observatories are probing the synchrotron emission of CR electrons and positrons.
Moreover, the era of neutrino astronomy was recently opened by the detection of 54 high-energy astrophysical events by the IceCube collaboration in four years of data taking [23]. Some of them are likely to be of Galactic origin as will be further investigated when the KM3NeT experiment [24] will be operating, providing a better coverage of both hemispheres. Such amount of recent and upcoming experiments has also another driver: the search for the first non-gravitational signal from Dark Matter (DM) particle interactions amid the
detected fluxes. DM indirect searches are strictly affected by uncertainties on the propagation of CRs in the Galaxy. Therefore, reducing and quantifying these astrophysical uncertainties is crucial in order to be able to disentangle a possible DM signature from astrophysical backgrounds and to determine the experimental sensitivity to DM properties.
In the wake of the plethora of novel experimental achievements described above, a parallel effort on the CR-transport modelling in the Galaxy is needed [25].
The DRAGON project has been pursued in order to meet the urgent demand to model CR propagation under the most realistic and general conditions.
Within this context, we introduce here the code version 2 (DRAGON2) as a general tool to simulate all relevant processes regarding CR transport from very low (∼ 10 MeV and below) to extremely high (∼ 1 PeV) energies.1
In particular, it computes the solution of the diffusion-advection-loss equation de-scribing CR transport for most CR species, from heavier ones down to protons, antipro-tons, and lepantipro-tons, both of astrophysical and exotic origin (i.e., coming from DM annihila-tions/decays). The transport equation features fully position- and energy-dependent trans-port coefficients (spatial and momentum diffusion, energy losses and advection) in both spatial two-dimensional (assuming cylindrical symmetry) and three-dimensional mode.
DRAGON2 allows a detailed study of both small-scale and large-scale structures (e.g., the spiral structure of the Galaxy) in steady-state and transient mode, refining the spatial resolution on the regions of interest (e.g., local bubble, GC, or Galactic Plane).
In this paper we introduce the code having in mind the specific case of CR propagation in the Galaxy; however, the code is written in a general way and can be easily used in many other different contexts and on different scales (e.g., CR transport in Galaxy clusters or in a star forming region).
Moreover, the new modular structure of the code make it possible for the user to im-plement additional spatial distributions for all relevant astrophysical quantities in a straight-forward way.
Old versions of DRAGON [26] have been used in several contexts. For example, to provide a solution to the CR gradient and isotropy problem in terms of inhomogeneous diffusion [27]; to compute CR electron and positron spectra in the presence of a spiral arm structure for sources [28]; to model CR antiprotons as CR secondaries and from DM annihilations [29,30]; to study the synchrotron emission from galactic leptons [31]; and to reproduce γ-ray and neutrino diffuse emissions above the TeV [32]. Most of these original results were obtained thanks to the innovative features already present in the first version of the code: these features are included also in the new version of the code and, where possible, extended to more general cases (see the discussion in Section 5).
With respect to the previous versions, the new code DRAGON2 was also largely reworked in order to optimally profit from modern programming design and computing techniques. In this paper, we provide a detailed description of the transport equation solver (see Section3) and provide in the appendixes details about different models for the relevant astrophysical ingredients (e.g., source and gas distribution, magnetic field models, spiral arm patterns) adopted (see Appendix C). In Section 4, we also provide a comprehensive set of numerical tests to assess the code performances, and to study the accuracy of the solution and the time needed to reach convergence in different conditions (e.g., different grid sizes, constant
1
At those energies, however, extrapolations of the spallation cross-sections must be performed and the source stochasticity may become relevant.
and variable time step). The main new features for galactic propagation are demonstrated in Section 5in a few example applications.
The first large project of this kind, GALPROP2, is a widely used code in the commu-nity [33–35]. GALPROP is designed to make predictions of direct CR measurements as well as gamma rays and synchrotron radiation consistently. It includes realistic models for nuclear spallation processes [36–40] and energy losses, but basic assumptions for the CR transport3. Semi-analytical solutions of the propagation equation are implemented in the USINE code developed since 2010 [41]. Taking advantage of much faster computation methods than nu-merical models, the semianalytical approach allows for an efficient scan of a wider transport parameter space [42–44].
Recently, the PICARD numerical code have been developed [45,46]. PICARD is fully 3D in concept and implements modern numerical techniques for the numerical solver, handling high resolutions with reasonable computer resources.
DRAGON2 is part of a complete suite of numerical tools designed to cover most of the relevant processes involving Galactic CRs and their secondary products over a very wide energy range. With the help of these tools – in particular the HeSky4 package – it is possible to compute spectra and sky-maps of radiation emitted by CRs interactions in a huge energy range, from the synchrotron radio waves up to the PeV neutrinos. On the low-energy side, the solar modulation can be treated either with auxiliary analytical routines implementing the force-field approximation [47, 48], or with the HelioProp numerical code featuring a detailed model of CR charge-dependent interaction with the Heliosphere, including diffusion, advection and energy losses due to the solar wind [49].
This paper does not contain a description of spallation processes and of off-diagonal anisotropic diffusion, which will be covered in forthcoming publications and in the evolving DRAGON manual (see www.dragonproject.org).
2 Transport of CRs in the Galaxy
DRAGON2 features all relevant processes for CR transport from Galactic acceleration sites to Earth: in particular, spatial and momentum diffusion, energy losses, advection, nuclear spallations and decays.
The combination of all these processes can be described by the following equation [50, 51]: ∇ · ( ~Ji− ~vwNi) + ∂ ∂p p2Dpp ∂ ∂p Ni p2 − ∂ ∂p h ˙ pNi− p 3 ~∇ · ~vw Ni i = Q +X i<j cβngasσj→i+ 1 γτj→i Nj − cβngasσi+ 1 γτi Ni (2.1)
where Ni(~r, p) is the density per total momentum p of the CR species i, Dpp(~r, p) is the momentum diffusion coefficient, Q(~r, p) describes the distribution and the energy spectra of sources, ~vw(~r) is the Galactic wind velocity responsible for CR advection, ˙p(~r, p) accounts for the momentum losses. The timescale for radioactive decay at rest is given by τi, while σi is
2Seehttp://galprop.stanford.eduandhttp://sourceforge.net/projects/galprop. 3
For a detailed comparison between the two codes we refer to the DRAGON2 wiki-page: https://github. com/cosmicrays/DRAGON2/wiki
the spallation cross-section with the interstellar gas. In this paper we do not consider these latter nuclear processes, and we postpone a detailed description to a forthcoming publication. The CR macroscopic current ~J (~r, p) is determined by the spatial diffusion tensor Dij, as Ji=−Dij∇jN .
These quantities can be either inferred from independent observations (e.g. the gas distribution, the magnetic field entering the loss term) or fitted to the data (e.g. the diffusion coefficient, the Galactic wind velocity). For all of them, different parameterizations are provided in literature and can be used to estimate the systematic uncertainty affecting the corresponding process. We therefore implement in DRAGON2 several options for the relevant transport quantities, as extensively described in Appendix C; in most cases, the quantities are position-dependent.
As discussed in the Introduction, one of the main novelty of our code with respect to other existing codes is the possibility to implement inhomogeneous transport5 (e.g., advec-tion, momentum and spatial diffusion).
In particular, assuming diffusion as inhomogeneous and anisotropic has a very natural motivation. In fact, the presence of a large scale Galactic magnetic field (GMF) clearly breaks isotropy and introduces a preferred direction, so that charged-particle diffusion should be expressed in terms of a diffusion tensor with components given by:
Dij = Dk− D⊥ bibj+ D⊥δij+ ijk DAbk, (2.2) where ~b is a unit vector along the mean (large scale) GMF. With this choice of versors, Dk and D⊥ are the components of the diffusion tensor parallel and perpendicular to the mean magnetic field and describe diffusion due to small-scale turbulent fluctuations. The coefficient DA gauges the anti-symmetric component of the diffusion tensor: It is usually identified as the drift coefficient since it describes a macroscopic drift orthogonal to both ~b and the gradient of the CR density, ~∇N [52, 53]. In this paper we always assume DA = 0 since the associated drifts are negligible up to ∼PeV energies as shown, e.g., in [54].
Although the physics behind CR diffusion is far from being understood (see e.g. [55] for a comprehensive review), some basic aspects may however be clarified starting from the weak-turbulence approximation where magnetic perturbations are well-developed in k-space and small compared with the regular background component. Under this assumption it is possible to treat analytically the problem of resonant CR interactions with the random-phase MHD wavemodes. This framework is known as quasi-linear theory (QLT) [56, 57]. The classical result for QLT gives that diffusion coefficients are described by a power-law in rigidity with different slopes for the parallel and perpendicular components (see also [58]). Moreover, these coefficients are spatially inhomogeneous since they are determined by local properties of the turbulent and regular fields. In this perspective, for the diffusion coefficients Dk and D⊥ we adopt several phenomenological parameterizations as proposed in recent works based on local fluxes and gamma-ray data (see AppendixC.8).
DRAGON2 can work either in a (2 + 1)-dimensional (2D) or in a (3 + 1)-dimensional (3D) configuration. In the 2D case we use cylindrical coordinates defined by the radial distance r and the height form the Galactic disk z and we assume azimuthally symmetry. For the 3D case we consider Cartesian coordinates x, y, z. The quantities defined as function of cylindrical coordinates are consistently mapped in Cartesian coordinates by the relation r =px2+ y2. In the next Sections, we will specify the transport equation in these two configurations.
2D in cylindrical coordinates and azimuthal symmetry
In 2D mode, Dk plays no role and the derivative of the particle density along the azimuthal coordinate vanishes (∂φN = 0), then Eq.2.1can be written by substituting:
~ ∇ · ~J → Drr(r, z, p) ∂2N ∂r2 + Dzz(r, z, p) ∂2N ∂z2 + χ(r, z, p) ∂N ∂r + ψ(r, z, p) ∂N ∂z (2.3) where: χ(r, z, p) = Drr(r, z, p) r + ∂Drr(r, z) ∂r ψ(r, z, p) = ∂Dzz(r, z, p) ∂z
At Galactic scale, it is common to assume an azimuthal mean field, ~B = B ˆφ, such that, as it follows from Eq. 2.2, Drr = Dzz = D⊥. In this configuration, the problem reduces to that of isotropic diffusion.
In presence of a vertical component of the mean field, as the case of the GC, with ~
B = B ˆz, we obtain Drr= D⊥ and Dzz = Dk and, in general, D⊥6= Dk. 3D in Cartesian coordinates
In this configuration Eq. 2.1can be written by substituting: ~ ∇ · ~J → Dxx ∂2N ∂x2 + Dyy ∂2N ∂y2 + Dzz ∂2N ∂z2 + + 2Dxy ∂N ∂x∂y + 2Dxz ∂N ∂x∂z + 2Dyz ∂N ∂y∂z + + ux ∂N ∂x + uy ∂N ∂y + uz ∂N ∂z (2.4) where ui =∇j Dij.
In the present work, we consider only the case in which off-diagonal components of the diffusion tensor, Di6=j, are null. Under this condition, the previous equation can be simplified as this: ~ ∇ · ~J → Dxx ∂2N ∂x2 + Dyy ∂2N ∂y2 + Dzz ∂2N ∂z2 + + ∂Dxx ∂x ∂N ∂x + ∂Dyy ∂y ∂N ∂y + ∂Dzz ∂z ∂N ∂z . (2.5)
3 Numerical solution of the transport equation 3.1 Discretization over grid
In order to solve the transport equation numerically it is necessary to discretise the equation, i.e. to write it on a discrete grid and transform derivative operators into finite differences. In cylindrical coordinates (2D) we consider a grid with two spatial coordinates (ri, zj) and one momentum coordinate (pq); the grid spacing is arbitrary and it may be irregular. In Cartesian coordinates (3D) the spatial grid is instead obtained with three coordinates: xi, yj, zk.
CR density at a given position, momentum, and time can be written on this lattice as Ni,j,qn or Ni,j,k,qn
where n is the time step index.
In order to replace the derivatives in the transport equation by their finite difference approximations, we mainly adopt the centred difference scheme for an irregular spaced grid:
∂N ∂x → Ni+1− Ni−1 xi+1− xi−1 ∂2N ∂x2 → 2 xi+1− xi−1 Ni+1− Ni xi+1− xi − Ni− Ni−1 xi− xi−1
which gives a truncation error O(∆x2) and O(∆x) for uniform and non-uniform grid respec-tively.
3.2 Iteration scheme
We rewrite2.1as a time-dependent equation and we find the stationary solution by evolving an initial condition (IC), N0
ijk, until it relaxes to an equilibrium solution, N ∞
ijk, for which the time derivative vanishes.
Schematically the transport equation can be now written as: ∂N
∂t =L(N) + Q (3.1)
where L is the operator which defines the transport equation. In its discretized version, Eq.3.1 becomes:
Nin+1− Nin
∆t = ˆLi+ Qi (3.2)
where ∆t is the time step, and i is now a unique index over the spatial-energy grid.
The algorithm we adopt to evolve the solution of the transport equation 2.1 at each time step is described in Sec. 3.3. The convergence criterion is introduced in Sec.4.5. 3.3 Local One Dimensional (LOD) operator splitting method
A well-known approach to find the solution of a diffusive-advection equation is the operator splitting method.
The basic idea of this algorithm is to consider the transport equation3.1as a linear sum of different evolution operators (e.g., radial diffusion, vertical advection, energy loss, ...):
∂N ∂t =
X
l
Ll(N ) + Q (3.3)
and for each of them to find a valid differencing scheme for updating N from timestep n to timestep n + 1, as the operator were the only one on the right-hand side of 3.1. The overall evolution in the time step ∆t is obtained by using separately all the operators in sequence.
This specific implementation of the method is known as Local One Dimensional (LOD) operator splitting. Clear advantages of this algorithm are that one can discretise indepen-dently the different operators using different methods and different boundary conditions. It also allows to have different time steps for the the different subproblems.
The transport equation in cylindrical coordinates (which is obtained by substituting2.3 for the CR flux divergence in2.1), can be conveniently written as the sum of 5 operators:
Lr = Drr ∂2N ∂r2 + Drr r + ∂Drr ∂r ∂N ∂r Lz = Dzz ∂2N ∂z2 + ∂Dzz ∂z ∂N ∂z La =− ∂(vwN ) ∂z Lp = ∂ ∂p p2Dpp ∂ ∂p N p2 Ll = ∂ ∂p ˙ pN −p 3 ∂vw ∂z N ≡ ∂p∂ h ˙P N i
where we assumed ~vw = vw(z)~z and vw(z) > 0, and ˙P ≡ ˙p − p3 ∂v∂zw. 3.4 Crank-Nicolson coefficients
The Crank-Nicolson (CN) method is a convenient discretisation scheme for ˆLi since it is second-order accurate in time and unconditionally stable. According to this scheme, the time derivative is obtained by taking the average of the right-hand side at t and t + ∆t, giving: ∂N ∂t =Ll(N ) + Q nos → Nin+1− Nin ∆t = 1 2 h ˆL n+1 i + ˆLni i + Qi nos (3.4) where nos is the number of active operators.
The most general expression of ˆL can be written as ˆ
Ln
i = LiNi−1n − CiNin+ UiNi+1n (3.5) and consequently 3.4 becomes:
−∆t2 LiNi−1n+1+ 1 +∆t 2 Ci Nin+1−∆t 2 UiN n+1 i+1 = ∆t 2 LiN n i−1+ 1− ∆t 2 Ci Nin+∆t 2 UiN n i+1+ ∆t nos Qi (3.6)
which is a tridiagonal set of simultaneous linear equations that we solve at each timestep to compute Nin+1 once Nin and Qi are given. Tridiagonal systems of linear equations can easily be solved by standard methods like Cholesky decomposition (see Sec. 2.9 in [59]) or LU decomposition (see Sec. 2.4 in [59]).
CN coefficients Li, Ci, and Ui, for the transport operators in the cylindrical symmetric version of the transport equation are reported in Table1. Boundary conditions (b.c.) are also reported in the same table: We assume N (r = R,|z| = H) = 0, and N to be symmetrical around r = 0. Boundary conditions in momentum are given by: N (p = pmax) = 0 and d dp N p2 pmin = 0 (see also B.4).
The three-dimensional anisotropic version in Cartesian coordinates is easily obtained by discretizingLx andLy similarly toLz with different diffusion coefficients (Dxx, Dyy, Dzz)
and by imposing as boundary conditions: N (|x| = R, |y| = R, |z| = H) = 0. The spatial operator CN coefficients in Cartesian three-dimensional grid are listed in Table2.
Differently than other operators, we adopt for the energy loss a 2-nd order accurate method which is obtained by integrating in momentum the energy loss term and evaluating the integral with the trapezoidal rule. This approach is described in detail in [45] and implemented in the PICARD code.
Following this idea, we end up with a tridiagonal set of linear equations:
(1 + Ci) Nin+1+ (1− Ui) Ni+1n+1= (1− Ci) Nin+ (1 + Ui) Ni+1n + ∆t (Qi+1+ Qi) (3.7) where Ci ∆t = − ˙ Pi pi+1− pi Ui ∆t = − ˙ Pi+1 pi+1− pi that we invert with standard methods to find Nin+1.
In Sec. 4.4, we compare the performances of this scheme with the 1-st order upwind discretisation combined with the CN scheme (see Table 1), as implemented in the previous version of DRAGON.
4 Validation of the numerical solver
In this section we discuss a complete set of numerical tests aimed at assessing the perfor-mances of the evolution equation components. In particular, our intent is to investigate both the convergence and the accuracy of the numerical approach that we use.
In order to test the convergence of the numerical solution, we introduce the concept of residuals. In a steady-state condition, the transport equation can be written as:
L(~x, p) + Q(~x, p) = 0 (4.1)
One can then introduce the normalised residual as:
R(~x, p) = L(~x, p) + Q(~x, p)Q(~x, p) (4.2) When solving the transport equation on a discretized grid, the quantityR corresponds to a matrix over the spatial-energy grid. To identify this matrix with one single value we compute the normalized l2-residual:
kRk2 = 1 NQ X i ˆLi+ Qi Qi !2 (4.3)
where NQ is the number of grid points where Q 6= 0. During the iteration procedure, the residual decreases and finally converges to a constant value which depends on the grid resolution and on the adopted discretisation scheme.
The tests that we discuss in this section are performed under simplified assumptions regarding both the geometry of the Galaxy and the astrophysical quantities. In doing so, we
0 2 4 6 8 10 t [Gyr] 10−7 10−6 10−5 10−4 10−3 10−2 k R k 2 n = 8 n = 16 n = 32 n = 64 n = 128 0 2 4 6 8 10 t [Gyr] 10−8 10−7 10−6 10−5 10−4 10−3 10−2 k R k 2 ∆t = 0.01 Myr ∆t = 0.05 Myr ∆t = 0.1 Myr
Figure 1. Left panel: the squared norm of the residualkRk2 as a function of the simulation time for
different spatial grid resolutions and for ∆t = 0.1 Myr. Right panel: the same for a fixed value of the grid resolution n = 64 and for different time steps.
are able to compare the numerical scheme of each transport operator against an analytical solution and this allows us to evaluate the accuracy of the numerical solution.
We quantify such accuracy by means of the relative error rel. This quantity is defined as follows: rel= max Na i − Nim Nia (4.4) where Nia is the analytical solution evaluated at the grid point corresponding to the index i, and Nim is the corresponding numerical solution as obtained after convergence has been attained (i.e., after the residual has reached the plateau value).
In the various tests, we evaluate the accuracy of the numerical solution for different values of the time step ∆t and of the grid spacings and we check if the scaling of rel with the grid step is consistent with the scaling expected from the discretisation order.
We derive the analytical solutions for the test cases used in this section in AppendixB. 4.1 Spatial diffusion
We first describe the case of a two-dimensional anisotropic and spatially dependent diffu-sion coefficient with a steady-state source term. The possibility to simulate inhomogeneous diffusion has been a relevant feature of the DRAGON code since its first version.
The transport equation adopted for this particular test, together with its analytical solution, is described in B.1. In particular, we study the case of anisotropic diffusion by setting f ≡ Dzz/Dxx= 0.1.
The left panel of Fig.1shows the evolution with time of the squared norm of the residual kRk2, for a fixed value of the timestep ∆t and for different spatial resolutions of the grid. The grid resolution is given in terms of the number of bins along r and z (we take nr= nz = n). The right panel of Fig. 1 illustrates the impact of the time-step on the residual, for a fixed spatial resolution.
0 2 4 6 8 10 r [kpc] 10−2 10−1 100 101 N [a.u.] ∆t = 0.1 Myr , n = 128 t = 0.1 Gyr t = 0.5 Gyr t = 10 Gyr analytical solution 100 101 102 103 n 10−7 10−6 10−5 10−4 10−3 10−2 max ((N − Na )/ Na ) × 0.01 × 0.1 ∆t = 0.01 Myr ∆t = 0.05 Myr ∆t = 0.1 Myr
Figure 2. Left panel: comparison between the numerical and the analytical solution at different times, for a fixed spatial resolution nr = nz = 128 (radial profile). Right panel: relative error as a
function of the spatial resolution for different timesteps.
We show in the left panel of Fig. 2 the comparison between the analytical and the numerical solutions at different times, for a fixed spatial and time resolution. In particular, the plot shows the profile along r (and for z = 0) of the two solutions. As it can be seen, the numerical solution reproduces remarkably well the analytical one.
The relative error as a function of the spatial resolution and for different timesteps ∆t is shown in right panel of Fig. 2. The relative error decreases proportionally to the grid step squared, as a result of the discretisation of the operators Lz and Lr being accurate up to the second order in r and z for a regular grid. We observe that the scaling remains valid down to a minimum resolution, ∆x ∼√Dxx∆t, below which the round-off error dominates the truncation one.6
4.2 Advection
We test the advection operator by assuming a Gaussian IC with σ = 500 pc and a constant (both in intensity and direction) advective wind along z with vw = 100 km/s. Differently than the other cases in which we compare with a steady-state solution, this is a typical Initial Value problem where a differential equation is given together with the unknown function in a given point of the solution domain (in this specific case t = 0).
The left panel of Fig. 3 shows the density profile at different times. As expected from the analytical solution (seeB.5), at each time the solution corresponds to the rigidly advected IC for ∆z = vwts, where tsis the simulation time. The maximum relative difference between the IC and the solution obtained at ts= 100 Myr is of ∼ 10−3 for ∆z = 20 pc and it exhibits a scaling with the grid size as given by the truncation order.
6It is a well-known fact (see, for example [60]) that values of ∆t/D
xx∆x2 larger than 1 can introduce
spurious oscillations in the numerical solution obtained with the Crank Nicolson scheme. In such condition, one does not expect the relative error to follow the scaling dictated by the discretization order.
−6 −4 −2 0 2 4 6 z [kpc] 0.0 0.2 0.4 0.6 0.8 1.0 N [a.u.] t = 0 t = 20 t = 40 t = 60 t = 80 t = 100 −4 −3 −2 −1 0 1 2 3 4 z [kpc] 10−3 10−2 10−1 100 101 102 (N − Na )/ Na CNi(z = 0) = 0 CNi(z = 0)6= 0 nz= 128 nz= 256 nz= 512 nz= 512
Figure 3. Left panel: solution of the advection equation at different times for vw = 100 km/s.
Right panel: relative difference as a function of the spatial resolution (red lines). Relative difference corresponding to the case with null CN coefficients at z = 0 (blue solid line).
A second test is performed to test specifically the case with discontinuous advective velocity at z = 0 as it is the case of a constant Galactic wind. The analytical solution is given by equation B.25.
We discretize the advective operator with a backward scheme for z > 0 and a forward scheme for z < 0. The CN coefficients at z = 0 are computed by summing the contribution of the z > 0 and z < 0 semi-intervals computed in the forward and backward schemes respectively (see tableA).
In the right panel of Fig. 3 we show the relative error of the numerical solution with respect to the analytical one for different grid sizes. The solution with nz = 512 points reproduces the analytical function better than 1%. In the same plot we compare the numerical solution obtained by assuming vanishing CN coefficients at z = 0 as implemented in the previous version of the code.
4.3 Momentum diffusion
We consider the reacceleration equation in B.13 with normalization of 5.1· 10−16 GeV2/s (vA = 50 km/s) and Q0 = 1. The corresponding analytical solution is obtained in B.4 and the discretization scheme is reported in table A.
Following the same strategy detailed in 4.1, we consider different values of ∆t and different grid sizes; the momentum interval is 0.1 ÷ 102 GeV, the number of intervals we consider is 32÷ 512. We show in Fig. 4 (left panel) the comparison between the numerical and the analytical solution for ∆t = 1 Myr.
We notice that the time needed to reach convergence is much larger at larger energies (> 105iterations in the case considered in the plot for p > 10 GeV). However, at momenta larger than ∼ 10 GeV the reacceleration operator is usually subdominant when spatial diffusion is also taken into account.
10−1 1 10 102 p [GeV ] 10−5 10−4 10−3 10−2 p 2 N [arbitrary units] ∆t = 1 Myr - np = 256 t = 50 Myr t = 100 Myr t = 500 Myr analytical solution 10 100 1000 np 10−3 10−2 10−1 1 (N − Na )/ Na ∆t = 1 Myr - p = 10 GeV
Figure 4. Left panel: comparison between the numerical and the analytical solution of the reac-celeration equation for ∆t = 1 Myr. Right panel: the relative error as a function of the number of momentum grid points for p = 10 GeV.
We show in Fig. 4 (right panel) the scaling of the relative error with the number of grid points. We obtain the ∝ n−1
p scaling as expected from the truncation order of the first derivative term in Lp.
4.4 Energy losses
The analytical solution of the energy-loss equation B.7 is shown in Sec. B.3.
We compare here the two methods presented in Sec. 3.4: a first-order Crank-Nicolson scheme as implemented in the previous version of the code, and a second-order more accurate scheme. Similarly as in the previous sections, we perform the numerical tests considering different values of ∆t and different grid sizes.
We compare in Fig. 5 the analytical solution with the numerical one as obtained for ∆t = 1 Myr with the first- and the second-order discretisation schemes. The reader can appreciate the better accuracy of the second-order approach. With np = 32 grid points in the given momentum range, the first-order scheme produces a ∼ 25% error with respect to the analytical solution, while the second-order scheme accounts for a ∼ 2.5% error only. We remark that this improvement in accuracy is obtained without a significant increase in the number of iterations required to reach convergence.
Absolute and the relative errors at p = 10 GeV as a function of the grid size is shown in Fig. 6. Expected scalings are correctly reproduced by the numerical solution. Remarkably, the second-order scheme provides a . 10−2 accuracy already with 64 grid points, while the same accuracy is reached by the first order scheme with more than 512 points.
4.5 Convergence and variable time step
In a realistic simulation, the code should be able to propagate particles in a huge range of energies and timescales.
10−1 1 10 102 p [GeV ] 10−11 10−10 10−9 10−8 10−7 10−6 p 2 N [a.u.] ∆t = 1 Myr - np= 32 - 1st order t = 10 Myr t = 100 Myr t = 1 Gyr analytical solution 10−1 1 10 102 p [GeV ] 10−11 10−10 10−9 10−8 10−7 10−6 p 2N [a.u.] ∆t = 1 Myr - np= 32 - 2nd order t = 10 Myr t = 100 Myr t = 1 Gyr analytical solution
Figure 5. Comparison between the numerical and analytical solution for ∆t = 1 Myr in both the first- (left panel ) and second-order (right panel ) cases, for energy loss term.
10 100 1000 np 10−15 10−14 10−13 10−12 10−11 N − Na ∆t = 1 Myr - p = 10 GeV first order second order 10 100 1000 np 10−5 10−4 10−3 10−2 10−1 1 (N − Na )/ Na ∆t = 1 Myr - p = 10 GeV first order second order
Figure 6. Absolute (left panel ) and relative error (right panel ) scaling with the number of grid points, for both second and first-order schemes.
As an example, spatial diffusion is determined by a diffusion coefficient whose depen-dence on the particle momentum is a power-law with slope δ∼ 0.3 ÷ 0.7 (see Appendix C.8). That implies that high-energy particles can diffuse much faster than the low-energy ones. In particular, if the range of energies under scrutiny spans several orders of magnitude the difference in the diffusion timescales can be quite large. As an example, with δ = 0.5, D0 ∼ 1.2 × 1028cm2s−1and an halo H = 4 kpc, a 10 TeV particle have a diffusion timescale of td= 2 Myr, which is 100 times faster than a 1 GeV particle.
100 101 102 103 104 p [GeV] 100 p 2. 7N [a.u.] ∆t = 1 - M = 31 ∆t = 1 - M = 100 ∆t = 1 - M = 310 ∆t = 1 - M = 1000 ∆t variable - M = 180 100 101 102 103 104 p [GeV] 10−5 10−4 10−3 10−2 10−1 relativ e difference m = 2 m = 5 m = 10 m = 20
Figure 7. Left panel: momentum spectra obtained by using a constant time-step ∆t (at different times) are shown in comparison with the solution found with a variable ∆t. Right panel: the relative difference with respect to the analytical solution is shown as a function of the number of iterations.
In terms of the numerical solution of the diffusion equation, the code must evolve with a time-step ∆t that has to be sufficiently small to correctly follow the diffusion of high-energy particles and it requires a time to reach convergence which is of the order of the longest timescale at the smallest energy. Because of this, the number of iterations that must be performed in order to find the correct numerical solution through all the energy spectrum could be extremely (and unnecessary) large.
An efficient algorithm to significantly reduce the computational effort has been proposed by [34] and a key feature of GALPROP. The basic idea is to change the value of ∆t within the single run, starting from a large time step and reducing it after a certain number of iterations is performed.
In the example we are considering here, we further assume source distribution having a Gaussian profile along z with characteristic scale 100 pc. We then start our simulation with ∆tmax= 100 Myr and we reduce the time step by a factor of 1.1 each m = 5 iterations, until ∆tmin= 1 Myr is reached. We remind from Sec. 4.1that the stability condition for diffusion reads: ∆t/td< 1, and we choose ∆t as half of the corresponding timescale.
We evaluate the performance of the code under this approach by comparing the local spectrum against the one we obtain by using a constant time-step with ∆tconst = ∆tmin. The results of the comparison are shown in Fig. 7.
As one can clearly see, by using a constant time-step, the numerical solution reproduces the analytical one only after ∼ 103 iterations. In fact, the use of a variable ∆t provides a very efficient way to reduce the number of iterations and an accurate numerical solution is found after 180 iterations only.
By showing the right panel of Fig. 7, we point out how the overall accuracy found by using a variable ∆t is affected by changing the value of m, namely the number of iterations for each time step. Since the total number of iterations is simply proportional to m, a trade-off between accuracy and computational time is required to choose this parameter.
10−1 100 101 102 103 p [GeV] 10−5 10−4 10−3 10−2 relativ e difference m = 2 m = 5 m = 10
Figure 8. The relative difference with respect to the analytical solution for the energy-loss equation is shown as a function of the number of iterations.
the energy loss normalisation, b0 = 10−16GeV/s, corresponding to the synchrotron energy-loss rate of an electron propagating in a ∼ 5 µG magnetic field. Under these conditions, the associated timescale, tl, ranges from∼ 3 Gyr at the lowest momentum to ∼ 0.3 Myr at 1 TeV. As in the previous example, we compare the case with a constant ∆tcst = tl,min/2 and a variable ∆t, starting from the largest timescale and decreasing until ∆tcstis reached. The figure 8, where we show the accuracy with respect to the analytical solution for different choices of the parameter m, confirms a similar behaviour as for the diffusion case.
In conclusion, we found that the variable ∆t scheme provides a large improvement in terms of number of iterations. As pointed out by [45], this procedure could however have a shortcoming. As shown in the above two examples, the numerical scheme parameters, as the largest and the smallest timestep, strongly depends on the physical parameters (e.g., diffusion coefficient, Alfv´en speed, etc.). Moreover, the same author discussed how changing the physical parameters in a GALPROP run could generate a final output far from convergence (see also the appendix A.1.1 in [61]). In fact, GALPROP solicits the user to tune the numerical parameters governing the time steps accordingly to the physical model chosen.
On the contrary, DRAGON2 code computes the minimum and maximum time steps ac-cording to the active operator timescales. This strategy ensures numerical convergence for most of the physical parameter space.
5 Physical applications based on new features
In this section we present the capabilities of the new features in DRAGON2 by showing a few example applications. These features are motivated by several pieces of evidence (partially discussed in the following sections) calling for a more realistic description of the CR transport in our Galaxy than the simplified one-zone model implemented in most of the literature.
This chapter is organised as follows.
• In Section5.1,5.2, and5.3we focus our attention on spatial-dependent and anisotropic diffusion, i.e., the most characterising novelties of DRAGON2. Besides being theoretically
motivated, the need for implementing these features comes from γ-ray observations, currently the most effective tracers of the CR distribution in the Milky Way. In par-ticular, we focus here on the so-called gradient problem and the slope problem.
The first anomaly – already noticed in pre-Fermi data (e.g., with COS-B [62]) – consists in a discrepancy between the radial profile of the ray emissivity (inferred by the γ-ray longitudinal profile) and the one computed with conventional propagation models, using an injection term based on pulsar or SNR catalogues: The former appears to be flatter along the Galactic plane at large Galactic radii. This problem was recently confirmed by Fermi-LAT observations [63,64]. Although several possible explanations have been proposed (e.g., a flatter CR source distribution in the outer Galaxy or a strong Galactic wind), this discrepancy may be the signature of a faster perpendicular escape through the halo in the inner region of the Galaxy [27].
The second anomaly is about the γ-ray spectrum: The comprehensive collection of GALPROP-based models provided by the Fermi-LAT collaboration in [65] underpredicts the data above 10 GeV in the inner Galactic plane; this discrepancy is present in all the considered setups, with different prescriptions for the gas and source distributions. This high-energy excess is the signature of a harder CR spectrum in the inner Galactic plane, as shown also in [15, 66, 67]; according to [68] the hardening can be explained by a progressively harder scaling of the diffusion coefficient in the inner Galaxy (see AppendixC.8).
• In Section 5.4, we investigate the consequences on the proton spectrum of including the hadronic energy losses due to pion production and we quantify the impact on the propagated proton slope.
• In Section 5.5we briefly discuss the possibility of propagating transient sources. This feature is relevant when CR injection cannot be assumed to be a continuous process in space and time. The imprint of a nearby recent SN event on the locally observed CR spectrum has been recently revised in [69] and [70]. Under peculiar circumstances (in particular, highly anisotropic diffusion along regular magnetic field lines connecting the source to the Solar System), the bulk of low-energy hadronic CRs observed locally could be the result of a single recent event. Motivated by these considerations, we include in our code the possibility to follow the propagation of CRs emitted by time-dependent sources, and present in detail a physical case with anisotropic diffusion.
• As mentioned in Section 3.1, one of the key features of the DRAGON2 code is the possi-bility to perform CR transport over a non-uniform spatial grid. This property proves useful whenever one wants to study a certain region of the Galaxy (e.g., the Galactic Centre) with much higher resolution than the rest of the domain. In Section 5.6 we present two illustrative examples in which the non-uniform grids can be convenient in terms of computational time.
• The relevance of reacceleration in CR propagation is still unknown: Local CR observ-ables cannot provide strong constraints on this process given the uncertainty on our understanding of∼ GeV diffusion. Since the new version of the DRAGON2 solver includes an alternative treatment of the boundary conditions in momentum, and a second-order scheme for the energy losses, it is interesting to compare the outcomes with what we would obtain by using the older solver. To this end, we describe the case of leptonic
2 4 6 8 10 12 14 16 18 20 r [kpc ] 0 0.2 0.4 0.6 0.8 1 N [a .u .]
Radial profile. Ferri`ere source term.
D = constant D∝ Q0.5
D∝ Q0.7
Figure 9. Radial profile of CR proton density along the Galactic plane, for three different choices of the parameter τ setting the dependence of D⊥ on the CR source density.
propagation, where the interplay of diffusion, energy losses (due to synchrotron emission and Inverse Compton), and reacceleration produces a peculiar feature in the spectrum at intermediate energies. We discuss this issue in Section 5.7.
5.1 Inhomogeneous diffusion coefficient: The gradient problem
Here we show how to solve the gradient problem in a position-dependent scenario with DRAGON2 working in 2D mode. DRAGON featured already inhomogeneous diffusion, this nu-merical test is shown to confirm one of the main results obtained with the previous version of the code.
Assuming cylindrical symmetry and a purely azimuthal structure of the regular GMF, only the perpendicular diffusion coefficient (D⊥) needs to be considered.
We assume D⊥ to be spatially correlated to the turbulence strength, hence to the CR source density q(r). Being the exact relation between those quantities poorly known, we parametrise it as in the PropToSourceTerm option described in Appendix C.8, with δ = 0.5 and η = 1. For the source term we choose the Ferriere2001 model, with a slope for momentum of 2.3.
We solve the 2D diffusion equation by adopting a variable time step ∆t, starting from the largest time step ∆t = 16 Myr and reducing it by a factor 0.5 each 100 iterations, until ∆t = 10−4 Myr is reached.
The main result is reported in Fig. 9: proton radial profile flattens, thus ameliorating the gradient problem, when the value of τ is increased (see also [27]).
We show further tests in Fig.10, where we focus on the model characterised by τ = 0.7. In left panel we show the effect of different spatial resolutions on the accuracy of the solution. A reference run with a very fine grid (∆r = 60 pc) is used as a comparison. It is interesting to notice that, due to the smooth distributions of all the relevant quantities, a
2 4 6 8 10 12 14 16 18 20 r [kpc] −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 relativ e difference ×10−2 D∝ Q0.7, p = 50 GeV ∆r = 125 pc ∆r = 250 pc ∆r = 500 pc ∆r = 1.0 kpc 5 10 15 20 25 30 r [kpc] 0 0.2 0.4 0.6 N [a.u.] D∝ Q0.7, p = 50 GeV, ∆r = 250 pc rmax= 15 kpc rmax= 20 kpc rmax= 30 kpc
Figure 10. Left panel: relative difference between the proton radial profiles computed with different spatial resolutions and with τ = 0.7. Right panel: proton radial profile as given by changing the radial boundary position.
percent-level accuracy all through the Galactic plane can be reached with a grid with ∆r = 1 kpc. In fact, we find that in most of the applications, a spacing of∼ 200 ÷ 500 pc is adequate to obtain an accurate solution, well below the level of 1%.
In right panel we consider the impact of changing the radial boundary condition. We find that the solution, in the region of interest (r < 10 kpc), is mildly dependent on the position of the boundary, and choosing a value for r larger than 15 kpc provide a negligible improvement in the accuracy of the solution.
5.2 Variable scaling of the diffusion coefficient
According to [68], the γ-ray longitudinal profile can be successfully reproduced by a progres-sively harder scaling of the diffusion coefficient in the inner Galaxy.
In DRAGON2, this scenario can be obtained by using the model VariableSlope for the diffusion coefficient (see AppendixC.8). In [68], a similar result was obtained with DRAGON by adding to the CN coefficients the additional terms containing the spatial derivatives of the rigidity power-law.
The spectra at different locations are shown in Fig.11. In left panel we reproduce the case as in [68] with a = 0.045 kpc−1 and b = 0.126 (giving δ(r ) = 0.5). These values provide a significant hardening in the inner Galaxy, compatible with the γ-ray data (see [68] for the details) .
The CR proton slope at different radii can be compared with the following prescription:
γ(r) = γinj+ δ(r), (5.1)
which would be the generalisation of the leaky-box model, and valid in the thin disk approx-imation.
10−1 1 10 102 103 p [GeV] 2 4 6 8 10 p 2. 8 N[a.u.] δ = 0.045r + 0.1265 r = 1.0 kpc r = 5.0 kpc r = 8.3 kpc 10−1 1 10 102 103 p [GeV] 10−1 1 p 2. 8N[a.u.] δ = 0.045r + 0.1265 r = 1.0 kpc r = 5.0 kpc r = 8.3 kpc
Figure 11. Left panel: CR proton spectrum on the Galactic plane at three different position as computed with a model with variable δ. Right panel: comparison of the proton spectra obtained with DRAGON2 (solid lines) with the power-laws given by Eq.5.1 (dashed lines). All curves are normalised to unit at 1 TeV.
In right panel of Fig.11we compare propagated proton spectra with those expected from Eq.5.1. In the specific case we are considering, the relation between injected and propagated slopes remains valid at large radii (in particular, it is obtained at the Sun position). As a consequence, a conventional propagation model tuned on local CR observables is still in agreement with local measurements (e.g., B/C) even if the spatially-dependent scaling is introduced.
5.3 Anisotropic diffusion coefficient
In this section we show how a radial dependence of the CR spectral index, similar to that discussed in the previous Section, may be obtained in a simplified framework of anisotropic diffusion.
Since CR diffusion is caused by their scattering off the magnetic field perturbations, and since such perturbations are usually expected to have components both in the perpendicular plane and along the parallel direction (with respect to the direction of the regular GMF), CRs can experience both a parallel and a perpendicular diffusion. As discussed in [58], numerical simulations of charged particle propagation in turbulent MFs found that these two components of the diffusion tensor are characterised by a different scalings with respect to the CR momentum.
In order to test such scenario, we consider a GMF with two components: • A purely azimuthal component, lying on the Galactic disk.
• An out-of-plane component, directed along the z-axis and confined within the bulge (R < 2.9 kpc).
0 5 10 15 20 r [kpc] 1029 1030 1031 diffusion co eff .[cm 2s − 1] D z Dr p = 1 TeV 100 101 102 103 104 p [GeV] 10−2 10−1 100 p 2. 8 N [a .u .] 0 kpc 3 kpc 8 kpc 17.9 kpc
Figure 12. Left panel: profiles of the diffusion coefficients along r and z for particles with p = 1T eV are shown. Right panel: energy spectrum computed at different radial distances from the Galactic Centre.
This model is a simplified version of the Jansson2012 model actually implemented in DRAGON2 and based on [71] (more details in AppendixC.2.1).
We expect therefore diffusion along the r-direction to be purely perpendicular, since the GMF has no radial component, while the diffusion coefficient along the z-axis is given by the sum of a parallel and a perpendicular term:
Dr= D0,⊥ p p0 δ⊥ Dz = D0,⊥ p p0 δ⊥ + D0,k exp −Rr 0 p p0 δk (5.2) with p0= 1 GeV.
For the normalization of the diffusion coefficients, we assume D0,k = 1028 cm2 s−1 and, following [58], D0,k/D0,⊥ = 30. As already said, parallel and perpendicular diffusion are characterised by a different dependence on particles momentum: following again the results found in [58], we assume that δk = 0.33 and δ⊥ = 0.5. The profile along r of the diffusion coefficients Dr and Dz for particles with p = 1 TeV is shown in the left panel of Fig. 12.
Concerning the geometry of the halo, we assume its radius to be 20 kpc, and its height to be 4 kpc. We assume as a source term a Gaussian disk with momentum power-law injection:
Q(p, r, z) = √1 2πzs exp −z 2 z2 s p p0 −2.3 . (5.3) with zs= 0.1 kpc.
We show in right panel of Fig.12the steady-state energy spectrum on the Galactic plane at different distances from the GC. For low values of R (within the bulge), parallel diffusion dominates and, as a consequence, a significant hardening can be noticed in the propagated
−4 −3 −2 −1 0 1 2 3 4 x [kpc] −4 −3 −2 −1 0 1 2 3 4 z [kp c] t = 0.1 Myr −4 −3 −2 −1 0 1 2 3 4 x [kpc] t = 0.5 Myr −4 −3 −2 −1 0 1 2 3 4 x [kpc] t = 1 Myr 10−15 10−14 10−13 10−12 10−11 10−10 10−9 10−8 10−7 10−6 CR flux at 1 GeV [cm − 2 s − 1 sr − 1 GeV − 1 ]
Figure 13. CR density in the x-z plane at different times; CRs are propagating anisotropically, with slower diffusion in the z direction.
slope; on the other hand, for larger values of R, the slope is steeper, and tends to 2.8, i.e. the value expected for perpendicular diffusion only.
5.4 Pion momentum losses
A novelty introduced in DRAGON2 with respect to earlier versions is a new implementation of pion production as a continuous loss term, in addition to ionisation and Coulomb losses.
In our numerical tests we consider a homogeneous source term confined in a disk (with scale height ' 100 pc); the energy-loss due to pion production relies on the analytical parametrisation reported in Section C.10.6, with the scale height zlosses= 100 pc.
Concerning the diffusion coefficient, we refer to the Eq.C.23, and implement a standard diffusive regime corresponding to D0 = 1.8· 1028cm2 s−1 at 1 GV, and δ = 0.5; The diffusive halo height is set to H = 4 kpc.
In Fig. 14we show the results of our numerical tests: We notice that the impact of the pion-production energy loss term on the proton flux ranges from ' 5% at ' 100 MeV down to few percent at energies larger than 100 GeV. The impact on the slope is . 0.5%.
5.5 Anisotropic diffusion from a transient source
The study of a transient source is relevant in many different context (e.g., to describe the Galactic centre activity). For this reason, we show here how DRAGON2 is able to follow the evolution of CRs emitted by an energetic source in a short event. Since the source is point-like and, in general, far from the centre of the coordinate system, we exploit the 3D mode; moreover, we consider anisotropic propagation, with Dr > Dz, inspired by a quasi-linear theory scenario dominated by parallel diffusion along the regular magnetic field line on the plane.
Our choice of the parameters is comparable with an average SNR event, as summarised here:
• kinetic energy released: 1051erg • efficiency of CR injection: 10%
10−1 100 101 102 p [GeV] −0.20 −0.15 −0.10 −0.05 0.00 0.05 0.10 relativ e difference diffusion π + diffusion ion. + diffusion
Figure 14. Relative difference on the proton spectrum with and without the pion-production and ionization energy loss terms and assuming D0= 1.8· 1028 cm2/s at 1 GV.
• CR injection spectrum: Φ = Φ0(p/p0)−2.3 in the range 0.1÷ 105 GeV • diffusion coefficient slope: δ = 0.5
• diffusion coefficient normalization on the plane: Dxx
0 = D
yy
0 = 1028 cm2/s at 1 GeV • perpendicular diffusion coefficient normalization: Dzz
0 = 1027 cm2/s at 1 GeV
The source is active from t = 0 to t = 5· 104 years. The evolution of the CR density in the x-z plane can be seen in Fig. 13: the signature of anisotropic diffusion is well clear in the central and right panel, corresponding to t = 0.5 and t = 1 Myr respectively.
In Fig.15 we show the evolution of the spectrum on the x axis, at a distance of 1 kpc from the source. For each rigidity the flux peaks after a time t compatible with the order-of-magnitude estimate given by D(p) = L2/(6t). The plot clearly shows the different timescales associated to CR diffusion at different rigidities, with the bulk of high-energy particles (' 100 GeV) arriving at the observer’s position after ∼ 0.5 Myr, and low-energy ones (' 100 MeV) after ∼ 10 Myr.
5.6 Non-equidistant binning Sources at the GC (3D mode)
We consider a three-dimensional source term given by: Q(x, y, z) = √1 2π 1 xsyszs exp − x 2 2x2 s − y2 2y2 s − z2 2z2 s (5.4) where xs= ys= 250 pc and zs = 100 pc represent the size of the source along the x, y and z axes, respectively. We consider a purely diffusive and isotropic scenario characterized by a diffusion coefficient defined as:
10−1 100 101 102 p [GeV] 10−5 10−4 10−3 p 2. 3N [GeV 1. 3cm − 2s − 1sr − 1]
Transient source. Distance = 1 kpc
t = 0.5 Myr t = 1 Myr t = 2 Myr t = 5 Myr t = 10 Myr t = 0.5 Myr t = 1 Myr t = 2 Myr t = 5 Myr t = 10 Myr
Figure 15. CR spectra at fixed distance and different times. The source is transient and is located at 1 kpc from the observer, on the Galactic plane. CRs diffuse anisotropically, with a larger coefficient along the plane and a lower one in the perpendicular direction.
Dxx,yy,zz(p) = D0 (5.5)
with D0= 1028 cm2 s−1.
We study the diffusion of CRs in this setup by performing a run characterised by a time-step ∆t = 0.01 Myr within a cube of edge 10 kpc. We find that, independently on the spatial grid used in the run, the numerical solution becomes constant after 2×104 iterations. We set the spatial resolution along the x and y axes to 156 pc (corresponding to 65 bins in each directions) and we test three different setups for the grid along the z axis:
• An Equidistant Binning (EB) with nz= 201, corresponding to a constant resolution of 50 pc from z =−5 kpc to z = 5 kpc.
• A Not Equidistant Binning (NEB) with nz = 27, where the bins width grows from 50 pc for|z| ≤ 100 pc to 100 pc in the interval 100 pc ≤ |z| ≤ 500 pc and then up to 1.36 kpc in the range 500 pc ≤ |z| ≤ 5 kpc (going to larger values of z, the width of each bin is larger than the previous one by a 50%).
• An EB with the same number of bins as the NEB setup that we have just described (27), corresponding to a spatial resolution of 385 pc.
The three different grids in the |z| ≤ 2 kpc region are depicted in left panel of Fig. 16. The results of the different runs are shown in the right panel of Fig. 16. In particular, the plot shows the CR density profile along the z axis, computed at x = y = 0 pc. As it can be seen, the numerical solution found with NEB is indistinguishable from the one given by EB with nz = 201. A crucial factor that has to be taken into account, however, is that in the former case the solution is found in 9222 seconds, while in the latter the run lasts for 1196 seconds before a stable solution is found. As for EB setup with nz = 27, the runtime is
−2.0 −1.0 0.0 1.0 2.0 z [kpc] nz=201 (EB) nz=27 (NEB) nz=27 (EB) −0.4 −0.2 0.0 0.2 0.4 z [kpc] 0.2 0.4 0.6 0.8 1.0 1.2 N [a.u.] nz= 201 (EB) nz= 27 (NEB) nz= 27 (EB)
Figure 16. In left panel the different setups for the binning of the z axis in|z| ≤ 2 kpc region are shown. Right panel shows the CR density profiles along the z axis that are obtained for such setups. Both panels refer to the uniform and isotropic diffusion coefficient example.
approximately the same as in the NEB case, but the solution found with such a binning can be slightly wrong, in particular in the peak region, where the error can reach 5%.
From these results, one could infer that in a scenario like the one that we considered in our test, which is characterised by a relatively spread CR distribution, using NEB determines a small increase in the accuracy of the numerical solution. Still this improvement comes at no computational cost, making the feature serviceable also in this case.
Sources on a disk (2D mode)
When particles are confined in a region that is much smaller than the size of the Galaxy, NEB can represent a very useful and necessary instrument to obtain a precise solution in a relatively short time. To provide an illustrative example of such a case, we consider a one-dimensional scenario in which we have a Gaussian source term defined as:
Q(z) = √1 2π 1 zs exp −z 2 z2 s (5.6) where zs = 100 pc. As before, we consider a purely diffusive case, but this time we assume that the diffusion coefficient drops by up to three orders of magnitude in the source region: D(z) = D0 1− 0.99 z 2 z2 s (5.7) where D0 = 1029cm2s−1. Under a physical point of view, one can motivate this decrease in the diffusion coefficient as a consequence of the stronger turbulence that characterises the region of the source.
We study the propagation of CRs in the -5 kpc ≤ z ≤ 5 kpc region for three different setups of the binning along the z-axis:
−1.0 −0.5 0.0 0.5 1.0 z [kpc] nz=501 (EB) nz=31 (NEB) nz=31 (EB) −0.4 −0.2 0.0 0.2 0.4 z [kpc] 0.6 0.7 0.8 0.9 1.0 1.1 1.2 N [a.u.] × 0.01 nz= 501 (EB) nz= 31 (NEB) nz= 31 (EB)
Figure 17. In left panel the different setups for the binning of the z axis in the|z| ≤ 2 kpc region are shown. Right panel shows the CR density profiles along the z axis that are obtained for such setups. Both panels refer to the non-uniform diffusion coefficient example.
• An EB with nz = 501, corresponding to a constant resolution of 20 pc.
• A NEB with nz = 31, where the bins width is 20 pc for|z| ≤ 100 pc and then grows to 50 pc, 100 pc, 500 pc and 1 kpc as larger values of z are considered.
• An EB with nz = 31, i.e. the same number of bins of the NEB setup described above. This number of bins corresponds to a spatial resolution of 333 pc.
Left panel of Fig. 17illustrates the binnings corresponding to the three setups described above in the region |z| ≤ 1 kpc.
The profile along the z-axis of the numerical solutions obtained for the three cases are shown in right panel of Fig.17. As it can be clearly seen, the solution that can be obtained with NEB overlaps perfectly with the one that is found in the EB case with nz = 501, while the solution that characterises the EB with nz = 501 appears to be wrong by more than two orders of magnitude. The advantage of using NEB is here evident, since by going from nz = 501 to nz = 31 the runtime decreases considerably (it goes from 46 seconds to 4 seconds), without any loss in the accuracy of the solution.
5.7 Interplay of diffusion, reacceleration, and leptonic energy losses: the role of the boundary condition in momentum.
The lepton spectrum is significantly shaped by the energy losses (∝ E2) due to synchrotron emission and inverse Compton scattering (see C.10 for the relevant formulas). We consider here a realistic setup in which the energy loss term is coupled with other operators: reaccel-eration and diffusion.
A significant reacceleration, in combination with ICS and synchrotron losses, produces a hump-like feature in the spectrum: In this section we aim at characterising this feature in a realistic setup.
10−2 10−1 100 101 102 p [GeV] 10−30 10−29 10−28 10−27 10−26 10−25 10−24 p 2. 5 N [a.u.] new solver old solver 10−1 100 101 102 p [GeV] −1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2 relativ e difference new solver old solver
Figure 18. Left panel: CR spectrum at the Sun position in arbitrary units. Red solid line: old solver (first-order energy loss, old prescription for the boundary condition); blue solid line: new solver (second-order energy loss, new prescription for the boundary condition). In both cases the CR injection is modelled as a power-law starting from pmin= 100 MeV (grey vertical line). Dashed lines
refer to the momentum boundary of the simulation set at 10 MeV. Right panel: relative difference with respect to the new solver, line legend as in the left panel.
As for the hadronic tests, we consider a homogeneous source term confined in a disk (with scale height ' 200 pc); the energy-loss term is taken as follows:
˙ p = exp −z 2 2zl · b0+ b2(p/p0)2 (5.8) with b0= 10−17GeV/s, b2 = 10−15 GeV/s, p0 = 1 GeV; the losses scale height is zl= 1 kpc. The diffusion coefficient is taken as in Eq. C.23, with D0 = 1028 cm2 /s and δ = 0.5; the Alfv´en velocity is 50 km/s. The diffusive halo height is H = 4 kpc.
We are interested in the [10 MeV – 100 GeV] momentum range; within this interval the energy loss timescale is [' 30 – ' 0.3 Myr], the reacceleration timescale is [' 6 – ' 600 Myr], the diffusion timescale is [' 6000 – ' 50 Myr]. The solution is obtained with the variable ∆t described in 4.5.
In Fig. 18 we show the CR spectrum at the Sun position, and we compare the new prescriptions for the discretisation of reacceleration and energy loss operators (described in detail in section 3) with the ones implemented in the first version of DRAGON.
The hump is clearly visible in the spectrum in both cases, and give rise to a peak at ' 2.5 GeV: slightly higher than the energy where the energy-loss and reacceleration timescales are the same (∼ 1 GeV). This feature is caused by the interplay between the two competing effects of energy loss (which is responsible for a downwards flux in momentum space) and reacceleration (which is a diffusive term in momentum space, and is responsible for an upwards flux due to the monotonically decreasing injection spectrum).
The DRAGON solver is characterised by a first-order discretisation for the energy losses, and the boundary conditions in momentum are obtained by imposing null derivatives for the
10−1 100 101 p [GeV] −0.10 −0.05 0.00 0.05 0.10 0.15 0.20 relativ e difference np= 256/dex np= 128/dex np= 32/dex np= 16/dex
Figure 19. The relative difference of the lepton spectrum for different energy grid sizes and the case np= 512/dex.
Crank-Nicolson coefficients. The DRAGON2 solver features a second-order discretisation for the energy losses, and as a boundary condition at the lowest momentum (see also section B.4):
∂ ∂p N p2 pmin = 0 (5.9)
The reference runs are shown in Fig.18: We set the minimum momentum of CR injection at pinjmin = 0.1 GeV and the boundary of the simulation at pmin = 0.1∗ pinjmin. In this setup, the two solvers exhibit a 30% difference at all energies due to the better accuracy of the second-order scheme.
We then compare each solver with the case in which pmin = pinjmin. We notice that the solution obtained with the new solver is practically unchanged by this prescription, while the solution of the DRAGON solver, with the old boundary condition, is significantly different at lower momenta. We conclude that the momentum boundary condition of DRAGON2 enables us to impose pmin= pinjmin with negligible effect on the solution accuracy.
In Fig.19, we consider a reference run with np= 512 points/dex and consider the impact of lowering the resolution. From this plot, we came to the conclusion that a resolution of np = 16 points/dex (i.e., 64 points in total in the momentum range where the source is injecting particles) yields an error as large as 15% in some portions of the spectrum. Even with a second order discretisation method, a ∼ 1% accuracy in the whole momentum range requires a much larger resolution (np ∼ 128 points/dex).
6 Conclusions
The simulation of galactic CR propagation plays an essential role in understanding the prop-erties of the galactic sources and of the interstellar medium. In this paper we introduced the new version of the publicly available galactic CR propagation code DRAGON2.
To interpret the recent measurements of primary CR fluxes and their secondary prod-ucts, the code was rewritten to solve the complete CR transport equation in 2D and 3D, for both leptonic and hadronic species, under very general assumptions.
The equation solver has been updated for all the operators: anisotropic and inhomoge-neous diffusion, advection, reacceleration, and energy losses. As discussed in the text, some of these features have been already exploited with modified version of DRAGON to study dif-fusion emissions and local spectra (see, e.g., [68,72]), that are now fully tested and included in the public code release. For each operator, we discussed the numerical scheme adopted in DRAGON2 and we tested against the corresponding analytical solution.
In doing so, we provide a complete set of analytical solutions that can be used as a benchmark test for any CR transport framework.
DRAGON2 features a wide collection of up-to-date models for all the relevant astrophysical ingredients involved in the computation: source term, magnetic field, interstellar radiation field, gas; all the distributions implemented in the code are presented in the paper.
We emphasised the novel features of DRAGON2 with respect to its previous version by simulating relevant physical cases where these features play a crucial role. In particular, we showed the implications of a different normalisation and rigidity scaling of the diffusion coef-ficient in different regions of the Galaxy, we described the case of CR anisotropic propagation from a transient source, we considered the possibility of a refinement of the grid in a peculiar interesting region, we discussed the interplay of diffusion, reacceleration and energy losses and the impact of these processes on the propagated leptonic spectrum.
DRAGON2 is an open source code under GNU general public license 7 distributed as a git repository (see www.dragonproject.org), and it can be easily upgraded or re-used to describe a wider range of physical conditions with respect to those treated in this paper.
Forthcoming publications will focus on additional aspects not covered in this paper, e.g., the general (non-diagonal) treatment of anisotropic diffusion, and a comprehensive de-scription of the cross-section network.
Acknowledgments
We are grateful to Luca Maccione who was one of the main authors and driving force of the DRAGON code and keeps providing us valuable advice and encouragement. We are also in-debted to Andy Strong for his support and many helpful discussions. We thank Iris Gebauer and her team (Matthias Weinreuter, Simon Kunz, Florian Keller) for their help with the debugging and testing of the solver, and for the implementation of the non-equidistant bin-ning. We warmly thank Richard Bartels, Francesca Calore, Elena Canovi, Silvio Sergio Cerri, Massimo Gaspari, Philipp Girichidis, Ralf Kissmann, Paolo Lipari, Giovanni Morlino, Mar-tin Pohl, Marco Taoso, Alfredo Urbano, Mauro Valli, Silvia Vernetto, for useful discussions, bug reporting, and feedback on the code. A.V. acknowledges the Gottfried Wilhelm Leibniz programme of the Deutsche Forschungsgemeinschaft (DFG) and a Grant from the GIF, the German-Israeli Foundation for Scientific Research.
We refer to our website www.dragonproject.org, where a short history of the DRAGON project is also reported, for a complete list of acknowledgements regarding routines and data extracted from external public codes.
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